# Part 3: Systems of autonomous ordinary differential equations, linear equations, and linearization of nonlinear equations --- 1. What is a two-by-two matrix, what is a column vector, how do we form products between matrices, what is the identity matrix, what is the determinant of a square matrix, and how can we identify eigenvalues and eigenvectors of a given matrix? 2. How do we convert linear systems of ODE into matrix form, what is the superposition principle, how do we define linear independence of solutions, and what is the fundamental solution set of a linear system of ODE? 3. How do we find the real general solution to two-by-two linear systems of autonomous and homogeneous ODE? 4. What is the relationship between vector fields and systems of ODE? How can we find equilibrium points of autonomous systems? How do we classify these equilibrium points? 5. What is the relationship between trace and determinant of a linear system and the corresponding equilibrium point classification? 6. How can we use the Jacobian as linear approximation to classify equilibrium points of nonlinear autonomous systems?  --- ### Part 3.1: Linear Autonomous Systems of Ordinary Differential Equations - Solution Forms in Two-Dimensions 1. Just Enough Linear Algebra to be a Danger to Yourself and Others * [The Matrix-Vector Product](https://youtu.be/95dyDM44_E8) (16 mins) * [02. Eigenproblems - Part 1 (definition, characteristic polynomial and vectors)](https://youtu.be/ZKjbhcooQGY) (9 mins) * [Review of homogeneous 2x2 matrix equations and relationship between determinant and linear independence](https://youtu.be/8_vJdcAEIrA) (8mins) * [Review of Eigenstuffs and calculation of eigenvalues/vectors for A=[0, 1; 1, 0]](https://youtu.be/iCP810Y2-C8) (17mins) + [Calculation of the second eigenvector of A=[0,1;1,0]](https://youtu.be/dTcAsrPcQR4)(7mins) * [Calculation of Eigenvalues/vectors of A=[3,-5; 5, -7] which is a repeated root case](https://youtu.be/IGEoYeub0vg) (7mins) + [Relationship between Eigenstuff and solutions to x' = Ax where A=[3,-5; 5,-7]](https://youtu.be/MonuXkwaE1M) (4mins) * [Calculation of Eigenvalues/vectors of A=[0,1;-1,0] which is an imaginary root case](https://youtu.be/sSRLI-ufIWk) (12mins) 2. Linear systems of homogeneous autonomous ODE * [Overview of the mathematical statements and the fundamental connection to eigenvalue/vector problems](https://youtu.be/zAptiDjVClA) (14mins) * [Wronskian determinant and the fundamental solution to x'=Ax where A=[0,1; 1, 0]](https://youtu.be/H0K5hSobcqE) (11mins) * [Getting sinusoid solutions to x'=Ax where A=[0,1; -1, 0]](https://youtu.be/Z1SM6Mb69Ns) (17mins) * [Recap and review of getting real-valued solutions in complex eigenvalue/vector cases](https://youtu.be/gh6W6s8OMAQ) (12mins) * [Solving the complex eigenvalue/vector problem x'=Ax, where A=[-3,-5; 3, 1]](https://youtu.be/Syao1C9LxFs) (11mins) * [Thinking about the repeated root case and highlighting the need for an additional vector, i.e., the generalized eigenvector](https://youtu.be/-3GEAeUQQV4) (12mins) * [Using the generalized eigenvector equation to solve x'=Ax, where A=[0, 1; -4, -4]](https://youtu.be/iH-JfgF7b64) (7mins) + [Generalized Eigenvector Full Discussion 1](https://youtu.be/F4Y6dGcySEM) (28mins) [F20 Resource with call back to 2019 bomb cyclone snow-day discussion) - [Bomb cyclone Equation for generalized](https://youtu.be/eDD0qfIqvac) (8mins) [S19 Resource created during a snow day due to the 2019 bomb cyclone] ### Part 3.2: Linear Autonomous Systems of Ordinary Differential Equations - Phase Plane Diagrams and the Trace Determinant Plane 1. Autonomous systems and equilibrium points * [Review of 2x2 linear autonomous results and defining equilibrium points in the phase plane](https://youtu.be/rnR-jlm94qM) (8mins) * [Finding equilibrium points in more complicated, i.e., nonlinear, autonomous systems](https://youtu.be/vRZ0_xXo9j8) (10mins) * [Converting a mass-spring system into a 2x2 system of linear autonomous ordinary differential equations](https://youtu.be/JnkElNkqhR4) (11mins) * [Representation of y'' + 5y' + 4y =0 as a system and initial discussions of the phase plane](https://youtu.be/UPml9-mw_F8) (12mins) 2. Phase Plane Diagrams * [Phase plane introduction with static and dynamic visualizations](https://youtu.be/wFeD5ZVc1zg) (9mins) * [Drawing a saddle whose eigenlines are orthogonal, A=[0,1;1,0], with visualization of dynamics](https://youtu.be/pR2_TKF3Du0) (18mins) + [Linear system with negative and positive eigenvalues (Saddle) and orthogonal eigenvectors](https://youtu.be/4mxjETCQuZo) (13mins) [F20 Resource w/ chapters] * [Drawing the phase diagram for an overdamped oscillator, y''+5y'+4y=0, with dynamic visualization](https://youtu.be/9u5hF7nLFVg) (13mins) + [Linear system with two negative eigenvalues (real sink) and non-orthogonal eigenvectors](https://youtu.be/py422LnAhMw) (20mins) [F20 Resource w/ chapters] * [Planar Autonomous Linear Dynamics: Real Source](https://youtu.be/7_MzLOhOTdA) (17mins) [F20 Resource w/ chapters] * [A discussion of phase plane diagrams to come via dynamic visualization](https://youtu.be/_zxWmL0B1n8) (4mins) * [Phase diagram of stable degenerate sink associated with A=[0,1;-4,-4] with dynamic visualization](https://youtu.be/Xd62pif6yNs) (14mins) + [Linear system with repeated negative eigenvalues (degenerate stable node)](https://youtu.be/ry62i3RJ1uA) (16mins) [F20 Resource w/ chapters] * [Summary of phase portraits for planar linear systems with real eigenvalues](https://youtu.be/7mKau58oedA) (16mins) [F20 Resource w/ chapters] * [Phase diagrams of neutrally stable centers and stable spiral sink with dynamic visualizations](https://youtu.be/DKYAMzVrFfs) (16mins) + [Solving y''+y=0 in vector form and finding the real sinusoid representation.](https://youtu.be/_qvoJzaJFM4) (20mins) [F20 Resource] + [Linear system with imaginary eigenvalues (center) consistent with y''+y=0](https://youtu.be/R6uSYdW5JI4) (10mins) [F20 Resource w/ chapters] + [Linear system with complex eigenvalues having negative real part(spiral stable sink)](https://www.youtube.com/watch?v=QhYxt6ko_1Q) (10mins) [F20 Resource w/ chapters] + [Linear system of autonomous ODE with complex roots: Finding the real-valued solution quickly.](https://youtu.be/iovdhUSjNAU) (6mins) [F20 Resource] * [Review of Planar Phase Diagrams and Connection to Mass-Spring Language](https://youtu.be/4znDDrfYevM) (1min) [F20 Resource] * [Phase diagram visualizations](https://www.evernote.com/shard/s352/sh/85d247c5-69a4-38fa-8543-1dd5bd9ebee9/F6L3gUCXr0IxL0Cg0j-NF2aTPo1tVfc7P3054LOwgsESZn1C9Iy97FXDZQ) + **Saddle Nodes**: Real different eigenvalues with different signs causing the intersection of a sink phase line with a source phase line - (0,0) is the equilibrium point of the system, [3, 2; 0, -3], and it is classified as a [saddle node](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0&cy=0&w=8.5398&h=8.5398&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%203.0*p.x%2B2.0*p.y%3B%0A%20%20v.y%20%3D%200.0*p.x-2.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D3%2C%20x1%20%3D%20%5B1%2C0%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B-2%2C5%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20saddle%20node%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%203.0*p.x%2B2.0*p.y%3B%0A%20%20v.y%20%3D%200.0*p.x-2.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D3%2C%20x1%20%3D%20%5B1%2C0%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B-2%2C5%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20saddle%20%0A%20%20return%20v%3B%0A%7D). - (0,0) is the equilibrium point of the system, [1, 3; 3, 1], and it is classified as a [saddle node](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0.000300000000000189&cy=0&w=8.5404&h=8.5404&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%201.0*p.x%2B3.0*p.y%3B%0A%20%20v.y%20%3D%203.0*p.x-2.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D3%2C%20x1%20%3D%20%5B1%2C0%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B-2%2C5%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20saddle%20node%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%201.0*p.x%2B3.0*p.y%3B%0A%20%20v.y%20%3D%200.0*p.x-2.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D3%2C%20x1%20%3D%20%5B1%2C0%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B-2%2C5%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20saddle%20node%0A%20%20return%20v%3B%0A%7D) - (0,0) is the equilibrium point of the system, [0, 1; 1, 0], and it is classified as a [saddle node](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0.0005999999999999339&cy=0&w=8.541&h=8.541&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%200.0*p.x%2B1.0*p.y%3B%0A%20%20v.y%20%3D%201.0*p.x-0.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D3%2C%20x1%20%3D%20%5B1%2C0%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B-2%2C5%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20saddle%20node%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%200.0*p.x%2B1.0*p.y%3B%0A%20%20v.y%20%3D%20-1.0*p.x-0.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D3%2C%20x1%20%3D%20%5B1%2C0%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B-2%2C5%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20saddle%20node%0A%20%20return%20v%3B%0A%7D) + **Real Stable Sinks**: Real different eigenvalues with negative sign, causing the intersection of two sink phase lines - (0,0) is the equilibrium point of the system, [-3, -1; -1, -3], and it is classified as a [real stable sink](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0&cy=0&w=8.5398&h=8.5398&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-3.0*p.x-1.0*p.y%3B%0A%20%20v.y%20%3D%20-1.0*p.x-3.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D-4%2C%20x1%20%3D%20%5B1%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B1%2C-1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Stable%20Real%20Sink%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-3.0*p.x-1.0*p.y%3B%0A%20%20v.y%20%3D%20-1.0*p.x-3.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D-4%2C%20x1%20%3D%20%5B1%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B1%2C-1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Stable%20%0A%20%20return%20v%3B%0A%7D). - (0,0) is the equilibrium point of the system, [-6, -1; 3, -2], and it is classified as a [real stable sink](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0.000300000000000189&cy=0&w=8.5404&h=8.5404&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-6.0*p.x-1.0*p.y%3B%0A%20%20v.y%20%3D%203.0*p.x-2.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D-4%2C%20x1%20%3D%20%5B1%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B1%2C-1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Stable%20Real%20Sink%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-6.0*p.x-1.0*p.y%3B%0A%20%20v.y%20%3D%203.0*p.x-3.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D-4%2C%20x1%20%3D%20%5B1%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-2%2C%20x2%3D%5B1%2C-1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Stable%20Real%20Sink%0A%20%20return%20v%3B%0A%7D). - [Prof. Nicolas video animation](https://youtu.be/OF70PpPfEyk) + **Real Unstable Sources**: Real different eigenvalues with positive sign, causing the intersection of two source phase lines - (0,0) is the equilibrium point of the system, [3, 1; 1, 3], and it is classified as a [real stable source.](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0&cy=0&w=8.5398&h=8.5398&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%203.0*p.x-1.0*p.y%3B%0A%20%20v.y%20%3D%20-1.0*p.x%2B3.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D2%2C%20x1%20%3D%20%5B1%2C1%5D%0A%20%20%2F%2F%20lambda2%3D4%2C%20x2%3D%5B1%2C-1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Unstable%20Real%20Source%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%203.0*p.x-1.0*p.y%3B%0A%20%20v.y%20%3D%20-1.0*p.x-3.0*p.y%3B%0A%20%20%2F%2F%20lamba1%3D2%2C%20x1%20%3D%20%5B1%2C1%5D%0A%20%20%2F%2F%20lambda2%3D4%2C%20x2%3D%5B1%2C-1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Unstable%20Real%20Source%0A%20%20return%20v%3B%0A%7D "real unstable source") + **Real Stable Degenerate Sinks**: Single real eigenvalue with negative sign, causing a single eigenline to draw trajectories to (0,0) - (0,0) is the equilibrium point of the system, [-3, 0; 1, -3], and it is classified as a [real repeated stable sink](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0&cy=0&w=8.5398&h=8.5398&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-3.0*p.x%2B0.0*p.y%3B%0A%20%20v.y%20%3D%201.0*p.x-3.0*p.y%3B%0A%2F%2F%20lamba1%3D-3%2C%20x1%20%3D%20%5B0%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-3%2C%20x2%3D%5B0%2C1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Unstable%20Real%20Source%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-3.0*p.x%2B0.0*p.y%3B%0A%20%20v.y%20%3D%201.0*p.x-3.0*p.y%3B%0A%2F%2F%20lamba1%3D-3%2C%20x1%20%3D%20%5B0%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-3%2C%20x2%3D%5B0%2C-1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Unstable%20Real%20Source%0A%20%20return%20v%3B%0A%7D) - (0,0) is the equilibrium point of the system, [0, 1; -4, -4], and it is classified as a [real repeated stable sink](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0.000300000000000189&cy=0&w=8.5404&h=8.5404&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-0.0*p.x%2B1.0*p.y%3B%0A%20%20v.y%20%3D%20-4.0*p.x-4.0*p.y%3B%0A%2F%2F%20lamba1%3D-3%2C%20x1%20%3D%20%5B0%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-3%2C%20x2%3D%5B0%2C1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Unstable%20Real%20Source%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-0.0*p.x%2B1.0*p.y%3B%0A%20%20v.y%20%3D%20-4.0*p.x-3.0*p.y%3B%0A%2F%2F%20lamba1%3D-3%2C%20x1%20%3D%20%5B0%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-3%2C%20x2%3D%5B0%2C1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Unstable%20Real%20Source%0A%20%20return%20v%3B%0A%7D) - (0,0) is the equilibrium point of the system, [0, 1; -9, -6], and it is classified as a [real repeated stable sink](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0.0005999999999999339&cy=0&w=8.541&h=8.541&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-0.0*p.x%2B1.0*p.y%3B%0A%20%20v.y%20%3D%20-9.0*p.x-6.0*p.y%3B%0A%2F%2F%20lamba1%3D-3%2C%20x1%20%3D%20%5B0%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-3%2C%20x2%3D%5B0%2C1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Unstable%20Real%20Source%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-0.0*p.x%2B1.0*p.y%3B%0A%20%20v.y%20%3D%20-9.0*p.x-.0*p.y%3B%0A%2F%2F%20lamba1%3D-3%2C%20x1%20%3D%20%5B0%2C1%5D%0A%20%20%2F%2F%20lambda2%3D-3%2C%20x2%3D%5B0%2C1%5D%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20Unstable%20Real%20Source%0A%20%20return%20v%3B%0A%7D) + **Imaginary Semi-Stable or Neutrally Stable Center**: Purely imaginary eigenvalues causing solutions to circulate about the (0,0) equilibrium solution - (0,0) is the equilibrium point of the system, [0, 6; -6, 0], and it is classified as a [center](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0&cy=0&w=8.5398&h=8.5398&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-0.0*p.x%2B6.0*p.y%3B%0A%20%20v.y%20%3D%20-6.0*p.x-0.0*p.y%3B%0A%2F%2F%20lamba1%3D%2B6i%2C%20%0A%20%20%2F%2F%20lambda2%3D-6i%2C%20%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20center%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-0.0*p.x%2B6.0*p.y%3B%0A%20%20v.y%20%3D%206.0*p.x-0.0*p.y%3B%0A%2F%2F%20lamba1%3D%2B6i%2C%20%0A%20%20%2F%2F%20lambda2%3D-6i%2C%20%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20center%0A%20%20return%20v%3B%0A%7D) - [Prof. Nicolas video animation](https://youtu.be/GSK5EGaDctI) + **Complex Stable Spiral Sink:** Complex eigenvalues with negative real part causing solutions to circulate about the (0,0) equilibrium solution as they spiral into it - (0,0) is the equilibrium point of the system, [-3, -5; 3, 1],  and it is classified as a [stable spiral sink](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0&cy=0&w=8.5398&h=8.5398&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-3.0*p.x-5.0*p.y%3B%0A%20%20v.y%20%3D%203.0*p.x%2B1.0*p.y%3B%0A%2F%2F%20lamba1%3D-1%2Bsqrt%2811%29i%2C%20%0A%20%20%2F%2F%20lambda2%3D-1-sqrt%2811%29i%2C%20%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20spiral%20sink%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%20-3.0*p.x-5.0*p.y%2B5.0%3B%0A%20%20v.y%20%3D%203.0*p.x%2B1.0*p.y%3B%0A%2F%2F%20lamba1%3D-1%2Bsqrt%2811%29i%2C%20%0A%20%20%2F%2F%20lambda2%3D-1-sqrt%2811%29i%2C%20%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20spiral%20sink%0A%20%20return%20v%3B%0A%7D). - [Prof. Nicolas video animation](https://www.youtube.com/watch?v=p5yU4mIObHc) + **Complex Unstable Spiral Source:** Complex eigenvalues with positive real part causing solutions to circulate about the (0,0) equilibrium solution as they spiral away from it - (0,0) is the equilibrium point of the system, [3, -5; 3, 1],  and it is classified as an [unstable spiral source](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0&cy=0&w=8.5398&h=8.5398&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%203.0*p.x-5.0*p.y%3B%0A%20%20v.y%20%3D%203.0*p.x%2B1.0*p.y%3B%0A%2F%2F%20lamba1%3D2%2Bsqrt%2814%29i%2C%20%0A%20%20%2F%2F%20lambda2%3D2-sqrt%2814%29i%2C%20%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20spiral%20source%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%203.0*p.x-5.0*p.y%3B%0A%20%20v.y%20%3D%203.0*p.x%2B1.0*p.y%3B%0A%2F%2F%20lamba1%3D2%2Bsqrt%2814%29i%2C%20%0A%20%20%2F%2F%20lambda2%3D2-sqrt%2814%29i%2C%20%0A%20%20%2F%2F%20EP%20%40%20%280%2C0%29%20is%20a%20spiral%20s%0A%20%20return%20v%3B%0A%7D) 1. Trace determinant plane * [Trace determinant plane - initial thoughts about relationship between eigenvalues and phase diagrams](https://youtu.be/w1oT-CqmIGs) (6mins) * [Trace determinant plane - summary of organizational structure and discussion of regions](https://youtu.be/3OmrhS29zVY) (16mins) + [The Trace Determinant Plane - Full Derivation w/ Time Stamps (chapters)](https://youtu.be/RstYksVUmdc) (30mins) [F20 Resource] - [Here](https://www.dropbox.com/s/8ejk1j23f9hvbi8/CatalogPhasePortraits.pdf?dl=0) you will find a catalog of phase space diagrams for linear systems. - [Here](https://www.youtube.com/watch?v=IakNOK4psmc) you will find a video recording of a software package used to navigate the trace determinant plane + [Practice classifying critical points using trace and determinant](https://youtu.be/GuIwirpLXN0) (16mins) [F20 Resource] * [Practice with classification via eigenvalues and the trace determinant plane](https://youtu.be/PhcRGbARG9U) (13mins) * [Visualization of walking through the trace determinant plane](https://youtu.be/yVw1HiWYz2Y) (7mins) [F20 Resource] ### Part 3.3: Nonlinear Linear Autonomous Systems of Ordinary Differential Equations - Linearization and Models 1. Linearization of a vector field using the Jacobian matrix * [The Jacobian matrix and Linearization of Planar Autonomous ODE](https://youtu.be/-f9y9pVireU)  (10mins) [F20 Resource] * [Review of the trace determinant plane](https://youtu.be/yIRGE5sFbMw) (3mins) 2. Competing species * [Discussion of competing species and calculating the autonomous system's equilibrium points](https://youtu.be/rOgiQW-2pYM) (14mins) + [Linearization of a competing species model (part 1) - Finding its equilibrium points](https://youtu.be/mcyuyRpG5qY) (9mins) [F20 Resource] * [Linear analysis of equilibrium points for competing species model with Jacobian and trace/determinant](https://youtu.be/oThoXykgHnw) (14mins) * [Linearization of a competing species model (part 2) - Classification of its equilibrium points](https://youtu.be/ThZUnn0m9Zc) (12mins) [F20 Resource] + Several equilibrium points for a competing species model [phase diagram](https://anvaka.github.io/fieldplay/?dt=0.01&fo=0.998&dp=0.009&cm=1&cx=0.6302000000000003&cy=2.11345&w=15.5868&h=15.5868&vf=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%202.0*p.x*%281.0-p.x%2F2.0%29-p.x*p.y%3B%0A%20%20v.y%20%3D%203.0*p.y*%281.0-p.y%2F3.0%29-2.0*p.x*p.y%3B%0A%0A%20%20return%20v%3B%0A%7D&code=%2F%2F%20p.x%20and%20p.y%20are%20current%20coordinates%0A%2F%2F%20v.x%20and%20v.y%20is%20a%20velocity%20at%20point%20p%0Avec2%20get_velocity%28vec2%20p%29%20%7B%0A%20%20vec2%20v%20%3D%20vec2%280.%2C%200.%29%3B%0A%0A%20%20%2F%2F%20change%20this%20to%20get%20a%20new%20vector%20field%0A%20%20v.x%20%3D%202.0*p.x*%281.0-p.x%2F2.0%29-p.x*p.y%3B%0A%20%20v.y%20%3D%203.0*p.y*%281.0-p.y%2F3.0%29-2.0*p.x*p.y%3B%0A%0A%20%20return%20v%3B%0A%7D "Link")